Research Teams

From equations arising from mechanics and physics, generally partial differential equations of elliptic or parabolic type, we aim to develop and implement a qualitative and quantitative study for this kind of problems. The study of these problems calls upon several tools of functional analysis. The abstract viewpoint adopted is essentially based on operator theory. The tools used include fractional calculus, singular perturbation methods, functional interpolation theory and linear operator semigroups.

As a second phase, we focus on the numerical approach. The implication of numerical methods such as finite differences, finite elements, finite volume, will aim to validate the theoretical results obtained previously.

Fractional derivative equations generalize integer-order derivative equations. The fractional approach appears today as a promising way to better describe real phenomena from finance, solid state physics, signal processing, quantum mechanics, porous media and many other fields. The development of fractional analysis itself represents a booming branch of mathematics. In this work, we propose to study some boundary value problems and fractional order derivatives. It is about providing sufficient conditions guaranteeing the existence of solutions and positive solutions and uniqueness of solutions. Our main tool will be the different techniques provided by fixed point theory.

Graph theory, which consists of studying the different properties of graphs, has become over the years a powerful tool of operations research for modeling and solving many practical problems (timetable problems, location problems, microprocessor architectures, etc.). Graph theory encompasses several areas, each as interesting as the others, including domination, coloring, stability, and kernels in directed graphs. A Graph Theory team will allow us to create the appropriate environment for research, deepen our knowledge and contribute.

Among the objectives of this team that we want to achieve, we mention:

• Contribution by original work in the field of graph theory
• Exchange, maintenance and acquisition from the scientific community of knowledge on the latest research developments on the stated themes
• Training of doctoral students within the framework of the new LMD system
• Creation of the research environment so that young people can flourish
• Development of software that makes it possible to answer certain real-world questions (industrial, economic and social) that can be formulated in terms of graph theory

The MEP research team focuses its activities both on theoretical and fundamental research axes and on practical subjects with immediate application identified and drawn from the real world in collaboration with the socio-economic sector. This research work concerns a range of questions related to classical queuing systems and/or with recalls to multiple servers taking into account the influence of server reliability and other parameters such as loss due to impatient customers, reserved times, passive failures and active failures and preventive maintenance times. These systems have many applications in telecommunications, computer systems and production systems.

The increasing complexity of computer systems makes the analysis, performance evaluation and guarantee of proper functioning increasingly necessary. In addition, the proliferation of new applications and services, new transmission media and new equipment places quality of service at the heart of the development of these systems. Performance evaluation becomes an essential task in order to verify quality of service constraints. However, the models of real systems are complex and very large. To this end, several evaluation techniques have been obtained, and several modeling methods are now used. Among these methods, we find stochastic comparisons, martingales as well as game theory techniques. Finally, experimental validation is essential in order to measure the complexity gains obtained in real cases.

The objective is to train researchers mastering the majority of modern algebra techniques, on both theoretical and practical levels. In particular, they will be able to model a concrete problem with algebraic models, give an idea of the difficulty of solving this problem and finally use and adapt recent fast algorithms to proceed with the resolution.

Research areas include:

• Elementary number theory
• Abstract algebra
• Witt rings
• Space operators
• Combinatorics

The main objective of our team is to model and predict catastrophic events. This modeling allows experts in the fields to create preventive scenarios to avoid human and economic damage to the country. This methodology finds an important place in several fields: hydrology, environment, economy, finance.

Our approach is based on a solid theory known as univariate and multivariate extreme value theory, the study of extreme events in cases of independence and dependence of events. Study of extreme data with the existence of censoring, applications to the estimation of risk measures. Study of conditional heavy-tailed distributions, estimation of the conditional quantile, quantile regression, applications to meteorological data to discover the impact of deterministic climate indices on a response variable (precipitation) in the Mediterranean area.

Study of extreme income distributions, application to the estimation of inequality indices in a capitalist population. Spatio-temporal modeling of precipitation by Max-stable processes, with an application to real data.

The scientific foundations of our team are based on several major themes of work in the field of fractional calculus. First, we focus on the in-depth study of the fundamental properties of fractional operators, in particular their asymptotic behavior, their stability and their regularity. Then, we explore the applications of fractional calculus in various fields, such as physics, engineering, biology and finance, with an emphasis on modeling and analyzing complex and nonlinear phenomena.

In addition, we develop new numerical and analytical methods to effectively solve differential and integral equations involving fractional operators. Finally, we are interested in the interdisciplinary aspect of fractional calculus by establishing links with other branches of mathematics and science, thus promoting the emergence of new synergies and research perspectives.